Remarks on Berry Connection in QFT, Anomalies, and Applications

TL;DR

This paper investigates the validity and properties of Berry connection in quantum field theories on compact spaces, exploring its relation to anomalies, boundary states, and applications to supersymmetric vacua and elliptic cohomology.

Contribution

It clarifies when Berry connection is well-defined in QFTs on compact spaces, relates it to boundary anomalies, and applies it to 3D theories and supersymmetric vacua.

Findings

Berry connection is well-defined in certain cases including tt* equations

Relation established between Berry connection and boundary anomalies

Application to 3D SUSY vacua and elliptic cohomology

Abstract

Berry connection has been recently generalized to higher-dimensional QFT, where it can be thought of as a topological term in the effective action for background couplings. Via the inflow, this term corresponds to the boundary anomaly in the space of couplings, another notion recently introduced in the literature. In this note we address the question of whether the old-fashioned Berry connection (for time-dependent couplings) still makes sense in a QFT on $\Sigma^{(d)}\times \mathbb{R}$, where $\Sigma^{(d)}$ is a $d$-dimensional compact space and $\mathbb{R}$ is time. Compactness of $\Sigma^{(d)}$ relieves us of the IR divergences, so we only have to address the UV issues. We describe a number of cases when the Berry connection is well defined (which includes the $tt^*$ equations), and when it is not. We also mention a relation to the boundary anomalies and boundary states on the Euclidean $\Sigma^{(d)} \times \mathbb{R}_{\geq 0}$. We then work out the examples of a free 3D Dirac fermion and a 3D $\mathcal{N}=2$ chiral multiplet. Finally, we consider 3D theories on $\mathbb{T}^2\times \mathbb{R}$, where the space $\mathbb{T}^2$ is a two-torus, and apply our machinery to clarify some aspects of the relation between 3D SUSY vacua and elliptic cohomology. We also comment on the generalization to higher genus.